Overlap Times in the $GI^B/GI/\infty$ Queue

TL;DR

This paper extends the analysis of overlap times in $GI^B/GI/ ightarrow ext{infinity}$ queues by incorporating batch arrivals, developing new measurement methods, and deriving exact results for multiple batch overlaps.

Contribution

It introduces two novel methods to measure overlap times with batch arrivals and provides exact analytical results for these scenarios, advancing understanding of customer interactions in such queues.

Findings

Developed two methods to measure overlap times with batch arrivals

Derived exact formulas for overlap times in infinite server queues with batch arrivals

Extended analysis to overlaps involving more than two batches

Abstract

Overlap times have been studied as a way of understanding the time of interaction between customers in a service facility. Most of the previous analysis relies on the single jump assumption for arrivals, which implies the queue increases by one for each arrival epoch. In this paper, we relax the single arrival assumption and explore the impact of having batch arrivals. Unfortunately, with batch arrivals it is not clear how one measures an overlap time between batches of customers. Thus, we develop two ways of capturing the notion of an overlap time in a batch setting and derive exact results in the infinite server queue with batch arrivals. Finally, we derive new results for analyzing overlap times of more than two batches.